Density of critical points for a Gaussian random function

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words, the density of the saddle points and extrema agree. This argument breaks down in three dimensions. All ratios of the densities of saddle points and extrema larger than one are possible. For a homogeneous Gaussian random field one finds no longer an equidistribution of signs, saddle points are slightly more frequent.Comment: 11 pages 1 figure, changes in list of references, corrected typo

One year of continuous measurements constraining methane emissions from the Baltic Sea to the atmosphere using a ship of opportunity

Methane and carbon dioxide were measured with an autonomous and continuous running system on a ferry line crossing the Baltic Sea on a 2–3 day interval from the Mecklenburg Bight to the Gulf of Finland in 2010. Surface methane saturations show great seasonal differences in shallow regions like the Mecklenburg Bight (103–507 %) compared to deeper regions like the Gotland Basin (96–161 %). The influence of controlling parameters like temperature, wind, mixing depth and processes like upwelling, mixing of the water column and sedimentary methane emissions on methane oversaturation and emission to the atmosphere are investigated. Upwelling was found to influence methane surface concentrations in the area of Gotland significantly during the summer period. In February 2010, an event of elevated methane concentrations in the surface water and water column of the Arkona Basin was observed, which could be linked to a wind-derived water level change as a potential triggering mechanism. The Baltic Sea is a source of methane to the atmosphere throughout the year, with highest fluxes occurring during the winter season. Stratification was found to promote the formation of a methane reservoir in deeper regions like Gulf of Finland or Bornholm Basin, which leads to long lasting elevated methane concentrations and enhanced methane fluxes, when mixed to the surface during mixed layer deepening in autumn and winter. Methane concentrations and fluxes from shallow regions like the Mecklenburg Bight are predominantly controlled by sedimentary production and consumption of methane, wind events and the change in temperature-dependent solubility of methane in the surface water. Methane fluxes vary significantly in shallow regions (e.g. Mecklenburg Bight) and regions with a temporal stratification (e.g. Bornholm Basin, Gulf of Finland). On the contrary, areas with a permanent stratification like the Gotland Basin show only small seasonal fluctuations in methane fluxes

Diagnostic, prognostic and predictive relevance of molecular markers in gliomas

The advances of genome-wide ‘discovery platforms’ and the increasing affordability of the analysis of significant sample sizes have led to the identification of novel mutations in brain tumours that became diagnostically and prognostically relevant. The development of mutation-specific antibodies has facilitated the introduction of these convenient biomarkers into most neuropathology laboratories and has changed our approach to brain tumour diagnostics. However, tissue diagnosis will remain an essential first step for the correct stratification for subsequent molecular tests, and the combined interpretation of the molecular and tissue diagnosis ideally remains with the neuropathologist. This overview will help our understanding of the pathobiology of common intrinsic brain tumours in adults and help guiding which molecular tests can supplement and refine the tissue diagnosis of the most common adult intrinsic brain tumours. This article will discuss the relevance of 1p/19q codeletions, IDH1/2 mutations, BRAF V600E and BRAF fusion mutations, more recently discovered mutations in ATRX, H3F3A, TERT, CIC and FUBP1, for diagnosis, prognostication and predictive testing. In a tumour-specific topic, the role of mitogen-activated protein kinase pathway mutations in the pathogenesis of pilocytic astrocytomas will be covered

Dynamical elastic bodies in Newtonian gravity

Well-posedness for the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Cosmological post-Newtonian expansions to arbitrary order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter \ep=v_T/c (0<\ep < \ep_0), where $c$ is the speed of light, and $v_T$ is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep \searrow 0 to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter \ep to any specified order with expansion coefficients that satisfy \ep-independent (nonlocal) symmetric hyperbolic equations

On the existence of dyons and dyonic black holes in Einstein-Yang-Mills theory

We study dyonic soliton and black hole solutions of the ${\mathfrak {su}}(2)$ Einstein-Yang-Mills equations in asymptotically anti-de Sitter space. We prove the existence of non-trivial dyonic soliton and black hole solutions in a neighbourhood of the trivial solution. For these solutions the magnetic gauge field function has no zeros and we conjecture that at least some of these non-trivial solutions will be stable. The global existence proof uses local existence results and a non-linear perturbation argument based on the (Banach space) implicit function theorem.Comment: 23 pages, 2 figures. Minor revisions; references adde